Suppose a boys birthday is 6 Iyar, which is on a date that all observe mourning practices in Sefiras Homer. For those that do not give a haircut to a child until the child is three, can they make an Upsherin on the birthday? If not should it be done before Pesach, Lag Ba'Omer or some other time? (sources)
Or letzion (3:17 3) writes that all children may have haircuts during sefirah since aveilus does not apply to children. However rav Elyashiv (hilchos chag bechag 7 fun 58) that it should be avoided unless there's a need like an upsherin
According to Arizal the upsherin is delayed to lag baomer.
See dirshu edition mishna beturah 493 note 18
The Rivevos Ephraim 1:336 and 2:155:19 talk about this. He seems to hold that for a mitzva its mutar ,however he makes a distinction for those who live in Eretz Yisroel who usually bring their son to meron on lag baomer(even some months earlier) .If he didn't do it on lab bomer then he should wait until after Tisha b'av(the tshuvua speaks about three weeks and sefira together) this seemingly the halacha fir people in Eretz Yistoel.For someone who lives in chutz laretz it seems that it would be mutar.See it inside(I skimmed through the tshuvos) .
Nit'e Gavriel has an entire chapter (chapter 12) devoted to this in the volume on תגלחת הילדים הולכתם לחדר וסעודת החומש. I can't paraphrase the whole thing here, but his conclusion is that "by law it's permitted to cut his hair, but the practice is to wait until lag baomer or to cut his hair the day before Pesach".
There Is An Age-old Custom To Give A Three Year Old Boy His First Haircut At The Age Of Three, And Also Have The Boy Start Donning A Head Covering, A Yarmulka, As Well As Tzitzis, At All Times.
This Ceremony, Referred To By Many As The "Upsherin" (Yiddish For Haircut) Is A Joyous Occasion As The Parents Can Begin Training The Child (Chinuch) In The Mitzvah Of Peiyos (The Sideburns Or Side Locks That Must Be Left, And Are Prohibited By The Torah To Totally Remove) As Well As Other Mitzvos.
This Is Even Deemed A Simcha To The Point Of Allowing This Haircut To Be Performed During, Or Even Postponed Until, Chol Hamoed.(See Shulchan Aruch Siman 531:6 And Sha'arei Teshuva Os 6) (From www.HalachaForToday.com)