The authority responsible for the cemetery should hold detailed records of all the plots allocated so contact Leiston Cemetery if that is were you think the burial took place - contact details available on the www.
From the Suffolk Burial Index also lists quite a few BEDWELL burials in the parishes surrounding Leiston as well - Kelsale, Rendham, Saxmundham.
The family shows up on the 1891C living in High Street, Leiston with Elizabeth's grandmother Charlotte Fisher c1816. Granddaughter Elizabeth Bedwell 1868, Grandson George Bedwell 1874 There are three other younger children all described as 'grandchildren' living at the address Rose Bedwell 1884, Lily Bedwell 1887 & Annie Bedwell 1890. There is also a lodger.
In 1851 Charlotte Fisher & husband John are living in Leiston with their family - Isaac 1840, Job 1842 & Jane 1845. Jane appears to marry a William Bedwell in the DecQ of 1865 (BMD Ref: 4a/1500 Blything) & in 1881 the family are living in Leiston with a son Walter 1867, George, 1874 & Elizabeth 1868.
Is it possible that this is your family? Elizabeth is almost certainly the daughter of William Bedwell & Jane (nee Fisher) but I am wondering if Rose, Lily & Annie were also their offspring as Elizabeth would have only been 16 when the youngest of these 3 children Rose was born. Do you have a copy of Lily's birth cert?
In 1891 William Bedwell is on the census lodging in Middleton, not far from Leiston, described as a widower, which could explain why his family are living with their maternal grandmother.
Hope this helps
K2
