How about let's not increase her Q mana cost. Let's not have another Lissandra.
How about we directly compare these two then, shall we?
Syndra mana: 250(+50)
Lissandra mana: 220(+50)
Syndra mana regen: 6.9 (+.6)
LIssandra mana egen: 5(+.4)
Syndra Q cost: 40/50/60/70/80
Lissandra Q cost: 85
Syndra Q damage: 70/110/150/190/230 (+60%)/ 264.5 (+69%) [at level 9, vs champions]
LIssandra Q damage: 75/110/145/180/215 (+65%)
Syndra Q cooldown: 4
Lissandra Q cooldown: 6/5.25/4.5/3.75/3
As you can see here, Syndra's Q mana cost is less than half that of lissandra's at low levels, scaling up to marginally more. Lissandra does get one free spell every 18 seconds (can be reduced by using spells) from her passive. On the other hand, Syndra's low level mana pool is slightly larger and her mana regen is universally higher. Her Q deals marginally less damage early, scaling up to do more, and has a lower cooldown early, although lissandra's scales up to be lower. Of course the two abilities are different in another way- lissandra's is a linear skillshot with 725-825 range (depending on if you hit something or not) with the potential to slow, and syndra's is a ground targeted aoe spell with 800 range, no cc involved. The final difference is that syndra's q has no cast time, while lissandra's does.
If you compare all of those differences, they seem like reasonably equal skills, fulfilling more or less a similar role on each champion. There is, however, one difference. Syndra's mana efficiency at low levels is marginally higher, and more importantly, her cooldown is lower. Let's take a case example. Let's say we both want to kill and enemy champion at level one, just using our Q. We'll suppose that they have an effective magical health of 700 (this factors in MR and mpen), and that both syndra and lissandra have 40 AP. They also both have +6 mana/5, from Dorans and utility masteries. This situation is a little ridiculous- no one is just going to stand there and take it, and you would obviously be using aas as well, but just to compare the two abilities, please bare with me. This scenario means it will take Syndra 8 qs to kill her opponent (assuming no hp regen), and Lissandra only 7. But let's look at those mana costs and cooldowns! Syndra's 8 Qs will require 28 seconds to cast and cost 320 mana. Lissandra's 7 Qs will require 36 seconds to cast and cost 595 mana (before factoring in her passive). Assuming all 7 of her Qs slow, Lissandra's 1st, 4th and 7th qs will be free, so the actual cost will be 340. Factoring in mana regen, Syndra can cast that many Qs and will have 56 mana left at the end. Lissandra cannot cast that many Qs. She will be out of mana prior to her 6th q, and will have to wait until her passive is ready before being able to cast it. She will have 40.9 mana at the time she would have used her 6th q. After waiting for her passive on the 6th Q, lissandra's mana will have regened enough to cast her 7th by the time it is off of cd, leaving her with 9.8 mana after 41 seconds.
What does that wall of text/math mean? Essentially it means that, even factoring in Lissandra's passive and mana costs, Syndra's Q is capable of outputting more damage than lissandra's. In fact, in the 41 seconds required for lissandra to use 7 Qs, Syndra is capable of using 10 (restricted by mana, not CD). This means that when both champs are out of mana at 41 seconds, Syndra will have output 940 damage (assuming all qs hit), and Lissandra will have output 707 damage (assuming the same). If we were to increase Syndra's Q mana cost at level 1 to 50, things would even out. Syndra could use 8 qs in the 41 seconds it takes lissandra to use 7, outputting only 40 more damage assuming they all hit.
Now, there's nothing in these numbers to say that this discrepancy in early game power is unjustified. I haven't compared their w, e, and r skills and so forth.
TL:DR Syndra is far, far stronger at low levels than lissandra, increasing her q cost by even 10 would mitigate this. This shouldn't be news, and this is really just a fact